Inertial Mann-Type Algorithm for a Nonexpansive Mapping to Solve Monotone Inclusion and Image Restoration Problems

In this article, we establish a new Mann-type method combining both inertial terms and errors to find a fixed point of a nonexpansive mapping in a Hilbert space. We show strong convergence of the iterate under some appropriate assumptions in order to find a solution to an investigative fixed point problem. For the virtue of the main theorem, it can be applied to an approximately zero point of the sum of three monotone operators. We compare the convergent performance of our proposed method, the Mann-type algorithm without both inertial terms and errors, and the Halpern-type algorithm in convex minimization problem with the constraint of a non-zero asymmetric linear transformation. Finally, we illustrate the functionality of the algorithm through numerical experiments addressing image restoration problems.

DYNAMIC RESIDUAL SEABED RESPONSE AROUND A MOVABLE PILE FOUNDATION

Wave-induced seabed soil response and its resultant liquefaction is common observed in a silt seabed with relative poor drainage condition, which poses a great threaten to the foundation safety of marine structures. Regarding the governing equations, three different approaches namely the Fully-dynamic (FD), Partialdynamic (PD) and Quasi-static (QS) model, have been used in the previous studies. Among these, both PD and FD approaches consider the effect of the inertial terms of soil skeleton/fluid. It has been reported in the literature that effects of the inertial terms on the seabed response could not be neglected, especially for the seabed around a movable structure (Ulker et al., 2010). However, these studies only focused on the oscillatory mechanism which are probably seen in a sandy seabed with high permeability. Recently, Zhao et al. (2017) investigated the residual soil response around a pile foundation by integrating a RANS wave model and a QS seabed model. In their study, the inertial terms of soil skeleton and pore water were neglected. To the authors’ best knowledge, up to now, effects of the inertial terms on the residual response of a silt seabed have not been investigated.